Burning Numbers of t-unicyclic Graphs

Abstract

Given a graph G, the burning number of G is the smallest integer k for which there are vertices \(x_1, x_2,\ldots ,x_k\) such that \((x_1,x_2,\ldots ,x_k)\) is a burning sequence of G. It has been shown that the graph burning problem is NP-complete, even for trees with maximum degree three, or linear forests. A t-unicyclic graph is a unicycle graph in which the unique vertex in the graph with degree greater than two has degree \( t + 2 \). In this paper, we first present the bounds for the burning number of t-unicyclic graphs, and then use the burning numbers of linear forests with at most three components to determine the burning number of all t-unicyclic graphs for \(t\le 2\).